**Problem : **

Write down the Gibbs sum. Be sure to get all of the indices correct.

**Problem : **

Give the expression for the absolute probability that a system will be
found in the state with
*N*
_{1}
particles and energy
.

**Problem : **

Give an expression for the average value of a property
*A*
for a system
in diffusive and thermal contact with a "reservoir". A "reservoir" is a
huge system next to our smaller system with large energy and number of
particles.

< *A* > =

**Problem : **

Give an expression for the average number of particles in a system that is in thermal and diffusive contact with a reservoir.

We are looking for
< *N* >
, which we can calculate using the formula we
just derived.

< *N* > =

**Problem : **

Suppose that we have a system that can be unoccupied or can have one particle in a state with energy . Write the Gibbs sum for this system.

One possible state has
*N* = 0
, for which we say that the energy
is also zero. So the first term in the sum is
1
.
The second possible state has
*N* = 1
, and energy
. We can
write the total sum as:

We sometimes simplify this by defining
*λ*âÉá*e*
^{
μ/τ
}
, in
which case the answer can be written more simply as
*Z*
_{G} = 1 + *λe*
^{-/τ
}
.